HW1 solution_09fall - 1.6(a Consider the following statistical model yi = li i i=1,2,6 where yi=the ith observed difference between A and B(A-B

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1.6 (a) Consider the following statistical model: i i i l y ε τ + + = , i =1,2,…,6, where y i =the i th observed difference between A and B (A-B) τ =the intrinsic difference between A and B (A-B) l i =learning effect of the i th transcript ε i =errors with mean 0 When the test sequence for the i th transcript is AB, B is benefited by the learning effect, thus l i <0. Similarly, l i >0 if the sequence of the i th test is BA. Assume that 0 ... 6 2 1 > = = = = l l l l in part (a). Without randomization, as the following sequence: AB, AB, AB, AB, AB, AB l y = = η ˆ , i.e., the estimation of the difference between A and B is biased by l . With randomization as the following sequence: AB, BA, AB, BA, AB, AB 3 6 2 4 ˆ l l l y = = = , i.e., the estimation of the difference between A and B is biased by l /3< l . Using balance in addition to randomization, as the following sequence AB, AB, AB, BA, BA, BA (*) = = = 6 3 3 ˆ l l y , i.e., there is no bias if we use balance. (b) I would not use the sequence (*). A better choice is as follows: AB, BA, BA, AB, AB, BA (**) For the sequence (*), all the 3 ABs are before all the BAs, which makes this sequence uneasy. Both sequence (*) and (**) do not cause estimation bias if l l l l = = = = 6 2 1 ... . However, for 0 ... 6 2 1 > > > > l l l , (**) leads to smaller bias than (*). 1.10 a)
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BodyWt BrainWt 7000 6000 5000 4000 3000 2000 1000 0 6000 5000 4000 3000 2000 1000 0 Scatterplot of BrainWt vs BodyWt The scatter plot above is not very informative. Due to the large values of Brain Weight and Body Weight for a few observations, the abscissa and ordinate axes have to cover very wide ranges. However, most of the observations are associated with small Brain Weight and Body Weight. Consequently, many points cluster around the bottom left corner of the scatter plot. BodyWt 600 500 400 300 200 100 0 800 700 600 500 400 300 200 100 0 Scatterplot of BrainWt vs BodyWt
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The above figure is a scatter plot of Brain Weight versus Body Weight with reduced range for the abscissa and ordinate axes. Note that several observations are omitted from the plot. However, it can now be seen that Brain Weight and Body Weight are positively correlated. b) Log(BodyWt) Log(BrainWt) 10.0 7.5 5.0 2.5 0.0 -2.5 -5.0 10 8 6 4 2 0 -2 Scatterplot of Log(BrainWt) vs Log(BodyWt) A scatter of the natural logarithm of Brain Weight versus the natural logarithm of Body Weight is shown above. It is clear from the plot that there is a linear relationship between the logarithm of Brain Weight and the logarithm of Body Weight. Taking the logarithm transformation on both variables has the remarkable effect of introducing almost perfect linearity in the relationship between the variables. c)
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This note was uploaded on 12/25/2011 for the course ISYE 6413 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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HW1 solution_09fall - 1.6(a Consider the following statistical model yi = li i i=1,2,6 where yi=the ith observed difference between A and B(A-B

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